We study homogeneous curves in generalized flag manifolds $G/K$ with $G_2$-type $t$-roots, which are geodesics with respect to each $G$-invariant metric on $G/K$. These curves are called equigeodesics. The tangent space of such flag manifolds splits into six isotropy summands, which are in one-to-one correspondence with $t$-roots. Also, these spaces are a generalization of the exceptional full flag manifold $G_2/T$. We give a characterization for structural equigeodesics for flag manifolds with $G_2$-type $t$-roots, and we give for each such flag manifold, a list of subspaces in which the vectors are structural equigeodesic vectors.
This paper provides a characterization of homogeneous curves on a geometric flag manifold which are geodesic with respect to any invariant metric. We call such curves homogeneous equigeodesics. We also characterize homogeneous equigeodesics whose associated Killing field is closed, hence, the corresponding geodesics is closed.
This paper provides a characterization and examples of homogeneous geodesics on full $G/T$ and $G_{2}$ flag manifolds. We discuss for generalized root systems the property of sum-zero triple of $T$-roots and give several applications of this result.
Let $U/K_Θ$ be a generalized flag manifold, where $K_Θ$ is the centralizer of a torus in $U$. We study $U$-invariant almost Hermitian structures on $U/K_Θ$. The classification of these structures are naturally related with the system $R_t$ of t-roots associated to $U/K_Θ$. We introduced the notion of connectedness by triples zero sum in a general set of linear functional and proved that t-roots are connected by triples zero sum. Using this property, the invariant G1 structures on $U/K_Θ$ are completely classified. We also study the Kähler form and classified the invariant quasi Kähler structur…
We investigate the orientability of a class of vector bundles over flag manifolds of real semi-simple Lie groups, which include the tangent bundle and also stable bundles of certain gradient flows. Closed formulas, in terms of roots, are provided.
It is well known that the Einstein equation on a Riemannian flag manifold $(G/K,g)$ reduces to a algebraic system, if $g$ is a $G$-invariant metric. In this paper we described this system for all flag manifolds of a classical Lie group. We also determined the number of isotropy summands for all of these spaces and proved certain properties of the set of t-roots for flag manifolds of type $B_n$, $C_n$ and $D_n$.
Let $G$ be a complex semi-simple Lie group and form its maximal flag manifold $\mathbb{F}=G/P=U/T$ where $P$ is a minimal parabolic subgroup, $U$ a compact real form and $T=U\cap P$ a maximal torus of $U$. The aim of this paper is to study invariant generalized complex structures on $\mathbb{F}$. We describe the invariant generalized almost complex structures on $\mathbb{F}$ and classify which one is integrable. The problem reduces to the study of invariant $4$-dimensional generalized almost complex structures restricted to each root space, and for integrability we analyse the Nijenhuis operat…
Let $P=G/K$ be a semisimple non-compact Riemannian symmetric space, where $G=I_0(P)$ and $K=G_p$ is the stabilizer of $p\in P$. Let $X$ be an orbit of the (isotropy) representation of $K$ on $T_p(P)$ ($X$ is called a real flag manifold). Let $K_0\subset K$ be the stabilizer of a maximal flat, totally geodesic submanifold of $P$ which contains $p$. We show that if all the simple root multiplicities of $G/K$ are at least 2 then $K_0$ is connected and the action of $K_0$ on $X$ is equivariantly formal. In the case when the multiplicities are equal and at least 2, we will give a purely geometric p…
Let $\mathbb{F}_{Θ}=G/P_{Θ}$ be a flag manifold associated to a non-compact real simple Lie group $G$ and the parabolic subgroup $% P_{Θ}$. This is a closed subgroup of $G$ determined by a subset $% Θ$ of simple restricted roots of $\mathfrak{g}=Lie(G)$. This paper computes the second de Rham cohomology group of $\mathbb{F}_Θ$. We prove that it zero in general, with some rare exceptions. When it is non-zero, we give a basis of $H^2(\mathbb{F}_Θ,\mathbb{R})$ through the Weil construction of closed 2-forms as characteristic classes of principal fiber bundles. The starting point is the computatio…
On the atomic decomposition of complete intersection in flag varieties
Institute of Mathematics and Informatics, Bulgarian Academy of Sciences（保加利亚科学院数学与信息学研究所）；Simons Center for Geometry and Physics, Stony Brook University（石溪大学西蒙斯几何与物理中心）；International Center for Mathematical Sciences (ICMS)（国际数学科学中心）；Instituto de Matemática, Estatística e Computação Científica (IMECC) da Universidade Estadual de Campinas (Unicamp)（坎皮纳斯州立大学数学、统计与科学计算研究所）
For a Kähler Manifold $M$, the "symplectic Dolbeault operators" are defined using the symplectic spinors and associated Dirac operators, in complete analogy to how the usual Dolbeault operators, $\bar\partial$ and $\bar\partial^*$, arise from Dirac operators on the canonical complex spinors on $M$. We give special attention to two special classes of Kähler manifolds: Riemann surfaces and flag manifolds ($G/T$ for $G$ a simply-connected compact semisimple Lie group and $T$ a maximal torus). In the case of flag manifolds, we work with the Hermitian structure induced by the Killing form and a cho…
In the elastic shape analysis approach to shape matching and object classification, plane curves are represented as points in an infinite-dimensional Riemannian manifold, wherein shape dissimilarity is measured by geodesic distance. A remarkable result of Younes, Michor, Shah and Mumford says that the space of closed planar shapes, endowed with a natural metric, is isometric to an infinite-dimensional Grassmann manifold via the so-called square root transform. This result facilitates efficient shape comparison by virtue of explicit descriptions of Grassmannian geodesics. In this paper, we exte…
